Based on an Oscar-winning Irish film starring Glen Hansard and Markéta Irglová, about a thirty-something Dublin busker who falls in love with a young Czech immigrant flower seller, the Broadway production took the top award for 'Best Musical'.
Its lead actor Steve Kazee was also honoured with the prize for 'Best Lead Actor in a Musical'.
[L: James Corden, R: 'Once' songwriters and film stars Glen Hansard and Markéta Irglová]
Neil Patrick Harris hosted the ceremony for the third time, wowing the Beacon Theater audience with an opening number based on the hit musical The Book of Mormon, which was the major winner at last year's ceremony.
"Welcome to the 66th Annual Tony Awards," he said, "Or, as we like to call it, 50 Shades of Gay."
Harris continued to impress throughout the night with several gags and comedic musical numbers as he introduced presenters including Jessica Chastain, Christopher Plummer, James Marsden and Tyler Perry.
There were several musical numbers throughout the night, with performances by Ricky Martin as Che in the revival of Evita, , 'Best Lead Actress in a Musical' winner Audra McDonald singing 'Summertime' from Porgy and Bess, and High School Musical's Corbin Bleu leading a number from Godspell.
Highlights of the night included appearances by Book of Mormon creators Matt Stone and Trey Parker, as well as James Corden performing scenes from his acclaimed show One Man, Two Guvnors, for which he won the 'Best Lead Actor in a Play' award later in the evening.
Corden was astonished by the win, saying during his acceptance speech: "I have to say... John Lithgow and James Earl Jones and Frank Langella - and my favourite actor in the world, Philip Seymour Hoffman - to be on a list with you was enough, and holding this honestly just reminds me there is no such thing as 'best'. Honestly, I am overwhelmed."
Les Misérables star Hugh Jackman was also presented with a Special Tony Award for his contribution to theater by his wife Deborra-Lee Furness.
Jackman, who has hosted the Tony Awards three times, joked that he thought producers were "on crack" when they first asked him to take on the role, but praised his colleagues for a warm welcome into the theatre community.
The show was closed by Harris, who performed a jazz medley recap of the two-hour ceremony, singing out the audience with a glass of whiskey in hand.